TAILIEUCHUNG - Báo cáo toán học: "A q-analogue of some binomial coefficient identities of Y. Sun"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: A q-analogue of some binomial coefficient identities of Y. Sun. | A q-analogue of some binomial coefficient identities of Y. Sun Victor J. W. Guo1 and Dan-Mei Yang2 Department of Mathematics East China Normal University Shanghai 200062 People s Republic of China 1jwguo@ 2plain_dan2004@ Submitted Dec 1 2010 Accepted Mar 24 2011 Published Mar 31 2011 Mathematics Subject Classifications 05A10 05A17 Abstract We give a q-analogue of some binomial coefficient identities of Y. Sun Electron. J. Combin. 17 2010 N20 as follows m n . n J q L 2J E k 0 Ln 4J E k 0 m k k q m 1 n 2k m k m 1 k q4 n 4k q n-22k q q n-24k q l 2 E 1 k k 0 m k m n 2k k q n 2k q where n q stands for the q-binomial coefficient. We provide two proofs one of which is combinatorial via partitions. 1 Introduction Using the Lagrange inversion formula Mansour and Sun 2 obtained the following two binomial coefficient identities L-Y 2 1 3k in k f 2k A M 3k J k 0 1 2n n 1 n - V Ợk A n k 7TT p n 1 . 2k 1 k 1 3k 1 n 1 n Ì v 7 v 7 1 H I 1 On 1 n T 1 lb k 0 x x z THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P78 1 In the same way Sun 3 derived the following binomial coefficient identities L 2J g k 0 1 3k a 3k a n a k k n 2k Ln 4J g k 0 Ln 4J g k 0 0 5k in k 42 1 k n k 2n 2 k n n n 4k 1 k 5k I n 1 k H n I I I 1 tv J otv J n 1 tv J n J k 0 L SJ 4 k0 n 1 2n a 2n ay n 4 1-3 1 1-4 k 0 n a 1 5k a n a k 4k a 1 y k y 5k a k 0 n a k 2n a 2k k n a h 1-5 It is not hard to see that both and are special cases of and is the a 0 case of . A bijective proof of and using binary trees and colored ternary trees has been given by Sun 3 himself. Using the same model Yan 4 presented an involutive proof of and answering a question of Sun. Multiplying both sides of by n a and letting m n a 1 we may write it as Ln 2J . J . I I Elm k m 1 m n k H n 2k I n n n k 0 while letting m n a we may write as Ln 4J z J Ln 2J z I J I Elm k m 1 _ X- kl m kwm n k n 4k 1 k m n m k 0 k 0 4 The purpose of this paper is to give a q-analogue of .

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